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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbv | Structured version Visualization version GIF version | ||
| Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44218. The equivalent of this theorem without the bound variable change is rexlimddv 3161. Usage of this theorem is discouraged because it depends on ax-13 2377, see rexlimddvcbvw 44219 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| rexlimddvcbv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | 
| rexlimddvcbv.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | 
| rexlimddvcbv.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexlimddvcbv | ⊢ (𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexlimddvcbv.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
| 2 | rexlimddvcbv.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
| 3 | rexlimddvcbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
| 4 | 2, 3 | rexlimdvaacbv 44218 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) | 
| 5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: (None) | 
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